However, as far I've seen, they can make doubles of themselves, stop time, shrink any object, explore alternate realities at will-- as far as this universe is concerned, that makes them omnipotent-they can do just about anything.

The federation can create doubles using the transporter, it happened accidentally so far but that can change, we also saw them shrinking a runabout by flying through a nebula, so the Q recreating something that happens naturally in the Star Trek universe isn't that impressive. Other things like time travel, alternate realities etc. could be illusions, Q claimed that it's all real but we only have his word.

Many things Q does are things the federation can do, only on a larger scale and without the use of obvious technology. Maybe super advanced Q technology is constantly scanning the galaxy and doing whatever a Q wants, instead of walking to the replicator, saying "Tea, Earl Grey, hot!" and carrying the cup back to your desk all a Q does is thinking, snapping his finger and the tea comes to him.

The Q claim they can do everything but is what we saw really that exceptional? The traveller sped up the Enterprise and it ended up god knows where, Q never even made them leave the galaxy. The Q blew up stars but so did Soran, the Q can "beam" and "replicate" but so can everyone else except the Kazon.

Q claimed that changing the gravitational constant of the universe is easy but we never saw him do anything on a universal scale, maybe he was just boasting. "I can fly, you can't but I conveniently lost my powers, so let's find another way to get the cat out of the tree".